Symbolic method (combinatorics): Difference between revisions

Revision as of 02:23, 24 October 2004

Revision as of 14:52, 6 December 2004

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===Other elementary constructions===

Other important elementary constructions are:

Other important elementary constructions are the ''cycle construction'' (<math>\mathfrak{C}\{\mathcal{B}\}</math>), which are like sequences except that cyclic rotations are not considered distinct, ''pointing'' (<math>\Theta\mathcal{B}</math>), in which each member of <math>\mathcal{B}</math> is augmented by a neutral (zero size) pointer to one of its atoms, and ''substitution'' (<math>\mathcal{B} \circ \mathcal{C}</math>), in which each atom in a member of <math>\mathcal{B}</math> is replaced by a member of <math>\mathcal{C}</math>.

*the ''cycle construction'' (<math>\mathfrak{C}\{\mathcal{B}\}</math>), like sequences except that cyclic rotations are not considered distinct

*''pointing'' (<math>\Theta\mathcal{B}</math>), in which each member of <math>\mathcal{B}</math> is augmented by a neutral (zero size) pointer to one of its atoms

*''substitution'' (<math>\mathcal{B} \circ \mathcal{C}</math>), in which each atom in a member of <math>\mathcal{B}</math> is replaced by a member of <math>\mathcal{C}</math>.

The derivations for these constructions are too complicated to show here. Here are the results:

{|

|-

|'''! Construction'''

|'''! Generating function'''

|-

|<math>\mathcal{A} = \mathfrak{C}\{\mathcal{B}\}</math>

|<math>A(z) = \sum_{k=1}^{\infty} \frac{\phi(k)}{k} \ln \frac{1}{1 - B(z^{k})}</math> (where <math>\phi(k)\,</math> is the [[Euler totient function]])

|-

|<math>\mathcal{A} = \Theta\mathcal{B}</math>

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|-

|<math>\mathcal{A} = \mathcal{B} \circ \mathcal{C}</math>

|<math>A(z) = B(C(z))\,</math>

|}